PHYSICAL REVIEW B 85, 245448 (2012)

Conductance fluctuations in graphene systems: The relevance of classical dynamics

Lei Ying,1 Liang Huang,1,2 Ying-Cheng Lai,1,3,4 and Celso Grebogi41School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA

2Institute of Computational Physics and Complex Systems, and Key Laboratory for Magnetism and Magnetic Materials of MOE,Lanzhou University, Lanzhou, Gansu 730000, China

3Department of Physics, Arizona State University, Tempe, Arizona 85287, USA4Institute for Complex Systems and Mathematical Biology, King’s College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom

(Received 23 April 2012; revised manuscript received 15 June 2012; published 28 June 2012)

Conductance fluctuations associated with transport through quantum-dot systems are currently understood todepend on the nature of the corresponding classical dynamics, i.e., integrable or chaotic. However, we find that ingraphene quantum-dot systems, when a magnetic field is present, signatures of classical dynamics can disappearand universal scaling behaviors emerge. In particular, as the Fermi energy or the magnetic flux is varied, bothregular oscillations and random fluctuations in the conductance can occur, with alternating transitions betweenthe two. By carrying out a detailed analysis of two types of integrable (hexagonal and square) and one type ofchaotic (stadium) graphene dot system, we uncover a universal scaling law among the critical Fermi energy, thecritical magnetic flux, and the dot size. We develop a physical theory based on the emergence of edge states andthe evolution of Landau levels (as in quantum Hall effect) to understand these experimentally testable behaviors.

DOI: 10.1103/PhysRevB.85.245448 PACS number(s): 72.80.Vp, 05.45.Mt, 73.23.−b, 73.63.−b

I. INTRODUCTION

A fundamental problem in quantum transport throughnanoscale devices is conductance fluctuations. Consider, forexample, a quantum-dot system. As the Fermi energy of theconducting electrons is varied, the conductance can exhibitfluctuations of distinct characteristics, depending on thegeometrical shape of the dot. Research in the past two decadeshas demonstrated that the nature of the corresponding classicaldynamics can play a key role in the conductance-fluctuationpattern.1–4 For example, when the classical scattering dynam-ics is integrable or has a mixed phase-space structure, therecan be sharp resonances in the conductance curve. However,when the classical dynamics is fully chaotic, the conductancevariations tend to be smoother.

There have been tremendous recent efforts in graphene5–8

due to its relativistic quantum physical properties and itspotential for applications in nanoscale electronic devices andcircuits. The study of transport in open graphene devicesis thus a problem of vast interest.8 For example, the roleplayed by disorder in conductance fluctuations in graphenewas investigated, where anomalously strong fluctuations9 orsuppression of the fluctuations10 were reported. A recent workhas revealed that, in graphene quantum dots, the characteristicsof conductance fluctuations also depend on the nature ofthe classical dynamics similar to those for conventional two-dimensional electron-gas (2DEG) quantum-dot systems.11 Inthese recent works, magnetic field is absent. The magneticproperties of graphene, however, are different from thoseassociated with 2DEG systems. For example, in graphene thequantum Hall effect can be observed even at room temperaturedue to the massless Dirac fermion nature of the quasiparticlesand significantly reduced scattering effects.12 Especially, thelinear energy-momentum relation13 in graphene stipulates thatthe Landau levels are distributed according to ±√

N , where N

is the Landau index, as opposed to the proportional dependenceon N in 2DEG systems.14

In this paper, we study conductance fluctuations in graphenequantum-dot systems in the presence of magnetic field. We

present two main results. First, in the parameter plane spannedby the perpendicular magnetic flux and the Fermi energy, thereare regions of regular and random conductance oscillations,respectively. As the Fermi energy or the magnetic flux ischanged, the fluctuations can be either regular or random,implying a kind of “coexistence” of regular and irregularconductance fluctuations as a single physical parameter isvaried. Second, an experimentally significant issue is howconductance fluctuations are affected by the size of thequantum dot in the presence of a perpendicular magneticfield. In a previous experimental study15 of quantum dotsof sizes ranging from 0.7 to 1.2 μm, the authors foundnearly periodic conductance oscillations as the magnetic-fieldstrength is varied. The frequency of the oscillation pattern, theso-called magnetic frequency, was found to follow a scalingrelation with the edge size of the dot.15 In a recent study ofthe magnetic scaling behavior in graphene quantum dots,16,17

it was found that for small dots of edge size less than 0.3 μm,the magnetic frequency exhibits a scaling relation with thedot area. Here we shall focus on an important set of scarredorbits and examine the resulting conductance oscillations. Wefind that, for graphene quantum dots, below the first Landaulevel, the conductance exhibits periodic oscillations with themagnetic flux and with the Fermi energy. In fact, the magneticfrequency scales linearly with the dot size. However, the energyfrequency, the inverse of the variation in the Fermi energy forthe conductance to complete one cycle of oscillation, scalesinversely with the dot size. Beyond the regime of periodicconductance oscillations, new sets of scarred orbits emergeand evolve as successive Landau levels are crossed, each withits own period, leading to random conductance fluctuations.The remarkable feature is that these scaling behaviors areindependent of the nature of the underlying classical dynamics,i.e., regular or chaotic. Considering that a large body of existingliterature points to the critical role played by the nature of theclassical dynamics in conductance fluctuations,1–4 our findingthat the presence of magnetic field can greatly suppress thissensitivity to classical dynamics is striking.

245448-11098-0121/2012/85(24)/245448(7) ©2012 American Physical Society

YING, HUANG, LAI, AND GREBOGI PHYSICAL REVIEW B 85, 245448 (2012)

The rest of the paper is organized as follows. Sec-tion II describes briefly the tight-binding Hamiltonian andthe nonequilibrium Green’s function method to calculate theconductance for graphene quantum dots. Extensive evidenceof periodic conductance oscillations and the emergence ofrandom conductance fluctuations is presented in Sec. III.In Sec. IV, we develop a theoretical understanding of thenumerical results based on the emergence of edge states andsemiclassical quantization. Conclusive remarks are presentedin Sec. V.

II. GRAPHENE QUANTUM DOTS ANDCONDUCTANCE CALCULATION

We use the standard tight-binding framework18 to computethe conductances through graphene quantum dots of variousgeometrical shapes, where pz orbitals and nearest-neighborhopping are assumed. The tight-binding Hamiltonian has theform

H =∑

i,j

−tij (c†i cj + H.c.), (1)

where the summation is over all nearest-neighbor pairs and c†i

(cj ) is the creation (annihilation) operator, tij is the hoppingenergy7 from atom j and to atom i, and the onsite energy hasbeen set as the reference energy as it is the same for all theatoms. In the absence of magnetic field, the nearest-neighborhopping energy is tij = t0 = 2.7 eV. When a perpendicularuniform magnetic field B with vector potential A = (−By,0,0)is applied, the hopping energy is altered by a phase factor:

tij = t0 exp(−i2πφi,j ), (2)

where φi,j = (1/φ0)∫ i

jA · dl, and φ0 = h/e = 4.136 ×

10−15 Tm2 is the magnetic flux quanta. For convenience, weuse magnetic flux through a hexagonal plaque of graphene,φ = BS, as a control parameter characterizing variationsin the magnetic-field strength, where S is the area of thehexagonal plaque composed of six carbon atoms. Thus, S0 =3√

3a20/2, where a0 = 1.42 A. Here, we treat graphene devices

as flat two-dimensional systems. Large ripples modify thehopping and can induce localization and additional transportfluctuations.19

At low temperature, the conductance G of a quantum-dotdevice is approximately proportional to transmission T andis given by the Landauer formula:20 G(E) = (2e2/h)TG(E).The standard nonequilibrium Green’s function (NEGF)method21,22 can be used to calculate the transmission, whichcan be expressed by18,23

T (E) = Tr(�LGD�RG†D), (3)

where GD is the Green’s function of the device given byGD = (EI − HD − �L − �R)−1, HD is the Hamiltonian ofthe closed device, the semi-infinite leads are accounted forby the self-energies �L and �R , and �L,R are the couplingmatrices given by

�L,R = i(�L,R − �†L,R). (4)

B

B

Magnetic field areaNone field area

xz

y

B

FIG. 1. (Color online) Schematic illustration of hexagonal-,square-, and stadium-shaped graphene quantum dots in a perpen-dicular magnetic field. Note that the magnetic field exists only in thedevice region.

The local density of states (LDS) for the device is

ρ = − 1

πIm[diag(GD)]. (5)

To be representative, we consider graphene quantum dotsof three different geometric shapes: hexagonal, square, andstadium, as shown in Fig. 1. Hexagonal geometry is interestingdue to the graphene lattice symmetry, i.e., the boundariesconsist of zigzag edges only. Thus, regardless of the devicesize, the boundaries remain to be zigzag. The square geometryhas both zigzag and armchair boundaries along the twoperpendicular directions, respectively. The classical dynamicsin these two structures are integrable. The stadium-shapedquantum dot, however, has chaotic dynamics in the classicallimit, which has been used as a paradigmatic system inthe quantum-chaos literature to explore various quantummanifestations of classical chaos.24

The geometrical parameters of the three types of devicesare as follows. For the hexagonal geometry the height (thedistance between the two parallel boundaries) is 10.934 nm.The width of the lead is 1.136 nm, which is chosen somewhatarbitrarily. For the square device the width is 10.934 nm andthe width of the lead is 1.136 nm so that the overall size iscomparable to the hexagonal dot. The size of the rectangularpart of the stadium dot structure is 16.898 ×10.988 nm and itslead width is 1.136 nm.

III. NEARLY PERIODIC CONDUCTANCE OSCILLATIONSAND EMERGENCE OF RANDOM CONDUCTANCE

FLUCTUATIONS

Figures 2–4 are representative examples of conductancevariations either with the Fermi energy for fixed magneticflux or with the magnetic flux for fixed Fermi energy, for thehexagonal, square, and stadium dot shape, respectively. In allcases, a critical point can be identified unequivocally (denotedby either E1 or φ1), where the conductance variations arenearly periodic on one side of the point and random on theother side. In particular, for all three geometrical shapes, forfixed magnetic flux, the conductance varies quite regularly forE < E1 but randomly for E > E1. For fixed Fermi energy,the conductance variations are regular for φ > φ1 and randomfor φ < φ1. Better insights into the transition from regularto random conductance variations (or vice versa) can begained by examining the typical LDS patterns about thecritical point. For example, for the hexagonal geometry, thereis a circularly localized pattern at E1 = 0.2350t , as shown

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CONDUCTANCE FLUCTUATIONS IN GRAPHENE SYSTEMS: . . . PHYSICAL REVIEW B 85, 245448 (2012)

0 0.1 0.2 0.3 0.4 0.5

0

1

G(E

)/G

0

E/t

φ=0.005φ0

0.002 0.006 0.01 0.014 0.018 0.0220

1

φ/φ

G(φ

)/G

0 E=0.35t

(a)

0

E1 E

2

φ1

φ2

φ3φ

4

(b)

E4E

3

FIG. 2. (Color online) Conductances of the hexagonal-shapedquantum dot. The height of the dot is WD = 10.934 nm and thelead width is WL = 1.136 nm. The device region contains 4158carbon atoms. (a) Conductance versus the Fermi energy EF for fixedmagnetic field φ = 0.005φ0. The energy values of the shown LDSpatterns are those of the Landau levels: E1 = 0.2350t , E2 = 0.3395t ,E3 = 0.4100t , and E4 = 0.4730t, respectively. (b) Conductanceversus the magnetic flux φ for fixed Fermi energy E = 0.35t .At this energy, there are Landau levels located at φ1 = 0.0115φ0,φ2 = 0.0057φ0, φ3 = 0.0037φ0, and φ4 = 0.0024φ0 (from large tosmall). The corresponding LDS patterns are also shown.

in Fig. 2(a), where the conductance of the dot structure iseffectively zero due to the localization of conducting electronsinside the device. Figure 2(a) also displays several similar,recurring LDS patterns at E2, E3, and E4. The ratios amongthese energy values are E1 : E2 : E3 : E4 = 1 : 1.44 : 1.74 :2.01 ≈ 1 :

√2 :

√3 : 2. We observe that the energy values are

approximately proportional to√

N , where N is the index ofEN . These behaviors have also been observed for the squareand stadium-shaped quantum dots. For example, Fig. 3(a)shows, for the square geometry, occurrences of the transitionbetween regular and random conductance fluctuations at E1 :E2 : E3 = 0.2344t : 0.3289t : 0.4021t ≈ 1 :

√2 :

√3 : 2 for

fixed magnetic flux 0.005φ0. The ratio is also consistentwith the Landau level distribution as in Eq. (6) below. InFig. 3(b), the Fermi energy is fixed at E = 0.4t , and thetransition points are φ1 : φ2 : φ3 = 0.01508φ0 : 0.00756φ0 :

0 0.09 0.18 0.27 0.36 0.45

0

1

G(E

)/G

0

E/t

φ=0.005φ0

0.004 0.012 0.02 0.0280

1

φ/φ

G(φ

)/G

0 E=0.4t

0

φ2

E1

E2 E

3

φ3

φ1

(a)

(b)

FIG. 3. (Color online) Conductance variations in a squaregraphene quantum dot of side length WD = 10.934 nm and lead widthWL = 1.136 nm, which contains 4802 atoms. (a) For fixed magneticflux, Landau levels are located at E1 = 0.2344t , E2 = 0.3289t , andE3 = 0.4021t , and so on. In (b) where the Fermi level is fixed, the tran-sition points are φ = 0.01508φ0, 0.00756φ0, 0.00501φ0, and so on.

0 0.1 0.2 0.3 0.4 0.5

0

1E/t

G(E

)/G

0

φ=0.0063φ0

0.002 0.006 0.010 0.0140

1

φ/φ

G(φ

)/G

0 E=0.3t

0

φ3 φ

1φ

2

E1

E2

E3

(b)

(a)

FIG. 4. (Color online) Conductance variations in the stadium ge-ometry. The rectangular region of this chaotic dot has the dimensionsWD = 16.898 nm and 10.988 nm, and the lead size is WL = 1.136 nm.The stadium shape contains 6410 atoms. Energy Landau levels arelocated at E1 = 0.263t , E2 = 0.369t , and E3 = 0.4471t , and soon for fixed magnetic flux. For fixed Fermi energy E = 0.3t , themagnetic Landau levels occur at φ = 0.0084φ0, 0.0042φ0, 0.0028φ0,and so on.

0.00501φ0 = 1 : 1/1.99 : 1/3.00, which are consistent withEq. (7) (to be discussed below). For the stadium-shaped device,the conductance curve shares the same features as Figs. 2 and 3.The transition points (as indicated in the figure and the caption)also fit into the same Landau level distribution as given byEqs. (6) and (7) below. These numerical results indicate that thecoexistence of regular and random conductance fluctuationsand the transitions between them are determined by the Landaulevels, regardless of the geometric shape of the graphenequantum dot. Note that, however, the LDS patterns do dependon the geometrical shape of the dot.

In nonrelativistic quantum, 2DEG systems of infinite size,the Landau levels are distributed linearly with the level indexN as EN = (N + 1/2)(eBh/m). However, for relativisticquantum quasiparticles in graphene, due to the linear energy-momentum relation E = vF k near the Dirac point, the Landaulevels are distributed according to25

E(N ) = ±ωc

√N, (6)

where ωc = √2vF /�B is the cyclotron frequency of Dirac

fermions (electrons) and �B = √h/eB is the magnetic length.

When a Landau level rises, the charge carriers are localizedapproximately at the center of the device, leading to a near-zero conductance. The numerically obtained LDS patternsthus indicate that the critical energy values, for example, inFig. 2(a), are nothing but the Landau levels.

From Eq. (6), we can obtain the corresponding Landaulevels in terms of the magnetic flux for fixed Fermi energy:

B(N ) = hE2

2ev2F

1

N. (7)

This formula can be verified by noting that, for example, asshown in Fig. 2(b), for fixed Fermi energy at E = 0.35t inthe hexagonal dot, varying the magnetic field also partitionsthe conductance curve into different regions with regular andrandom conductance fluctuations. The critical magnetic fluxesare φ1 = 0.0115φ0, φ2 = 0.0057φ0, φ3 = 0.0037φ0, and

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YING, HUANG, LAI, AND GREBOGI PHYSICAL REVIEW B 85, 245448 (2012)

φ4 = 0.0024φ0, leading to the approximate ratios of 1 : 1/2 :1/3 : 1/4, which is consistent with Eq. (7).

IV. SEMICLASSICAL THEORY OF REGULARCONDUCTANCE OSCILLATIONS AND UNIVERSAL

TRANSITION TO RANDOM CONDUCTANCEFLUCTUATIONS

Our numerical computations indicate strongly that theemergence and properties of the Landau levels are key tounderstanding the origin of regular conductance oscillationsin the presence of magnetic field. In fact, significant physicalinsights can be gained from the phenomenon of integerquantum Hall effect in semiconductor 2DEG systems, whichis a direct manifestation of the evolution of the Landau levels.In that case, when the magnetic field strength is fixed and theFermi energy is increased, the conductance reaches minimumwhen the Fermi energy is at a Landau level and takes on amuch larger value when the Fermi energy is in between twoneighboring Landau levels. This is contrary to the behaviorof the density of the states, which is appreciable only at theLandau levels. The basic reason is that, for a quantum dot,at the Landau level the charge carriers tend to be localizedin the central region of the dot and so cannot participate inthe transport process. However, when the Fermi energy is inbetween two adjacent Landau levels, edge states arise whichcirculate around the boundary of the quantum dot, facilitatinga strong coupling with the propagating modes in the semi-infinite leads and resulting in a large conductance. In our case,there is a new feature. Between two neighboring Landau levels,the energy difference Eh, where the subscript “h” standsfor Hall effect, is enormous so that, besides the formationof the circular edge states associated with the quantum Halleffect, another class of circular edge states can be formed, asstipulated by the semiclassical Bohr-Sommerfield quantizationcondition. This introduces another energy period Eq , where“q” stands for quantization, in which the Bohr-Sommerfieldedge states form and disappear. Since the circular edge statesfacilitate transport through the quantum dot and since Eq istypically smaller than Eh, the fulfillment of the semiclassicalquantization condition contributes to fine-scale oscillations inthe conductance curve.

To exploit the Bohr-Sommerfield quantization conditionfor the edge states in graphene, it is convenient to modify thesize of the device but keep the geometric shape unchanged.Without loss of generality, we focus on the hexagonal geometrythat possesses zigzag boundaries. We choose (somewhatarbitrarily) several heights of the hexagonal devices: WD1 =19.454 nm, WD2 = 10.934 nm, and WD3 = 6.674 nm withthe relative ratio WD1 : WD2 : WD3 = 2.9 : 1.7 : 1. Figure 5shows, for these devices, periodic conductance oscillationsbelow the first Landau level.

Bohr-Sommerfield quantization theory stipulates that theaction integral for two successive edge states satisfies thecondition26 I = h, where h is the Planck constant andI = ∮

p · dq. In the presence of a magnetic field with vectorpotential A, the generalized momentum is p = hk + eA andthe wave vector k has the same direction as dq. For a givenperiodic orbit of length L, we have

I = |p|L = h|k|L + eBS, (8)

0.1 0.12 0.14 0.16 0.18 0.20

1

G(E

,φ=0

.006

3φ0)/

G0

(a) Δ Eq=0.013t

0.1 0.15 0.2 0.25 0.30

1

G(E

,φ=0

.011

4φ0)/

G0

(b) Δ Eq=0.023t

0.1 0.2 0.3 0.40

1

G(E

,φ=0

.022

8φ0)/

G0

E/t

(c) Δ Eq=0.038t

0.014 0.016 0.018 0.02 0.0220

1

φ/φ

G(φ

,E=0

.3t)

/G0

0.004 0.0045 0.005 0.0055 0.0060

1

G(φ

,E=0

.15t

)/G

0

Δφq=0.0002φ

0

0.005 0.007 0.009 0.0110

1

G(φ

,E=0

.2t)

/G0

(d)

(e) Δφq=0.0007φ

0

(f) Δφq=0.0018φ

0

0

FIG. 5. (Color online) Conductance oscillations in hexagonalquantum dots of different sizes. The device width for (a) and(d) is WD = 19.454 nm and it contains 12938 atoms, for (b) and(e) it is WD = 10.934 nm and the device has 4158 atoms. In (c) and(f), the device has width WD3 = 6.674 nm and 1616 atoms. Everysubfigure indicates the period of the regular oscillations.

where S is the area that the periodic orbit encloses in thephysical space. For a fixed magnetic-field strength, we thenhave kL = 2π , where L is length of the periodic orbit.For graphene, we have E = hvF k near the Dirac point, sothe relationship between the energy interval Eq due to thequantization condition and the length of the periodic orbit is

Eq = hvF /L. (9)

Due to the different boundary conditions in two dimensions,we only test the ratio of the energy interval. In Figs. 5(a), 5(c),and 5(e), the energy intervals can be determined, givingthe ratios Eq1 : Eq2 : Eq3 = 1/L1 : 1/L2 : 1/L3 = 1 :1.76 : 2.92, which are quite close to the inverse ratiosof the device size 1/WD1 : 1/WD2 : 1/WD3 = 1 : 1.7 : 2.9.Moreover, for L = WD , we can estimate the Fermi velocityvF = EqWD/h ≈ 106 m/s, which is close to the Fermivelocity calculated from the dispersion curve. This meansthat the length of the circulating orbit is comparable to thedevice height, indicating that the effective diameter of theorbit is smaller than that of the device. We thus see thatthe regular conductance oscillations are a consequence of theBohr-Sommerfield quantization of the edge states between twoLandau levels. In particular, when the quantization conditionis satisfied, a strong LDS pattern emerges at the edge of thedevice, as shown in Fig. 6, which bridges with the transmittingmodes in the two leads and leads to the peak value 2e2/h

for the conductance. On the contrary, when the quantizationcondition is violated, edge states cannot form, giving rise tominimal conductance. Similarly, for fixed Fermi energy, orequivalently, fixed wave-vector (from the dispersion relation),the quantization condition becomes (eBS) = h, or

φ = BS = φ0, (10)

where φ0 = h/e is the magnetic flux quanta. Since theedge states typically circulate the device boundaries, S is

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CONDUCTANCE FLUCTUATIONS IN GRAPHENE SYSTEMS: . . . PHYSICAL REVIEW B 85, 245448 (2012)

E/t

φ/φ 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.004

0.008

0.012

0.016

0.02

0

0.5

1

1.5

2

2.5

3

21

FIG. 6. (Color online) A hexagonal geometry device with 4158atoms. Colors of the contour lines represent conductance G/G0. Thered dash lines are four Landau-levels, which divide the G ∼ E ∼φ contour into several distinct regions. In each region, there areone or several conductance fluctuation patterns (patterns 1,2,3). Eachconductance pattern corresponds a distinct LDS pattern.

proportional to the area of the device. From Figs. 5(b), 5(d),and 5(f), we obtain φq1 : φq2 : φq3 = 1 : 3.2 : 9 ≈1/W 2

D1 : 1/W 2D2 : 1/W 2

D3. Compared with the numerical re-sults of Eq , the error in φq is larger due to our approx-imation of S. When the area surrounded by the circulatingorbit is determined more precisely, we find that the magneticquantization condition [Eq. (10)] is satisfied. Note that in theabsence of magnetic field or if the field is weak, edge statesoccur only at zigzag boundaries. However, under a strongmagnetic field (above the first Landau level), edge states canemerge for both armchair and zigzag boundaries.

From the above analysis of the Bohr-Sommerfield quanti-zation condition, we find that the conductance oscillations arerelated to the Fermi energy, the magnetic-field strength, and thesize of the device. To obtain a quantitative scaling relationshipamong those parameters, we develop the following physicalanalysis. Theoretically, the size of a device is related to theelectron cyclotron radius at the Fermi energy, because onlythe electrons near the Fermi surface contribute to devicetransmission or conductance. The ratio of the cyclotronsurrounding area and perimeter is given by17

S/L = kF �2B, (11)

where S/L can be regarded as a single parameter character-izing the device size. In a graphene system, the energy neara Dirac point is proportional to the Fermi wave-vector kF :EF = hvF kF or kF = EF /(hvF ), where the Fermi velocityis given by vF = √

3t0a/2h and a = 2.46 A is the graphenelattice constant. Substituting these back into Eq. (11), weobtain the relationship of Fermi energy E, the device sizeD, and the magnetic flux φ as follows:

S/L = 2hS0√3eat0

E

φ, (12)

or in a different form as (for a given, fixed device size)

S/L = 2hS0√3eat0

E

φ. (13)

This relation can be used to infer the characteristic size D ofthe device from the conductance oscillations. For example, forthe hexagonal geometry, S = √

3D2/2 and L = 2√

3D. Thescaling relation can be modified to

Dhex = 12hS0√3eat0

E

φ, (14)

which can be readily verified numerically. In particular, sincethe curves shown in Fig. 5 are for the edge states circulatingthe device, we can use E and φ from the figure to infer thecorresponding values of D, which yields D1 = 13.926 nm,D2 = 7.836 nm, and D3 = 4.734 nm. Comparing with theactual size of the dot WD as described in the caption of Fig. 5,we observe somewhat large discrepancies. However, if wecompare the ratios, we have D1 : D2 : D3 = 2.94 : 1.655 : 1,which are extremely close to the ratios of the actual dot sizesWD1 : WD2 : WD3 = 2.915 : 1.640 : 1. We also see that, forthe three dot sizes, the ratio WD/D is the same, which isabout 1.4. The discrepancies in the actual size are caused bythe approximation in Eq. (11) and by the assumption that thediameter of the circulating orbits is equal to the device size.Nevertheless, since the estimated values of D and WD are ofthe same order of magnitude, it can be used to infer the dot sizefrom the conductance oscillations versus the Fermi energy andthe magnetic flux, which can be used as corroborative evidenceand be compared with other direct/indirect measurements.

The scaling relation (14) may be feasibly observed ex-perimentally in graphene quantum dots because, for lowFermi energy, the underlying phenomenon emerges evenwhen the applied magnetic field is weak, i.e., φ → 0. Forconventional semiconductor 2DEG systems with a parabolicenergy-momentum relation, similar scaling can in principle beobserved but only for enormous magnetic field, as we haveverified numerically. In particular, for a graphene quantumdot of size D ∼ 1 μm, the minimally required magnetic-fieldstrength to observe the periodic conductance oscillations isabout 3T . While for a 2DEG device of the same size as 1 μmmade of GaAs/AlGaAs heterogeneous structure, the minimummagnetic field required is27 about 10T .

To obtain a global view of the conductance oscilla-tions/fluctuations in terms of a combination of Eqs. (6) and(7), we overlay the Landau levels on top of the contour plotof the conductance versus both energy E and magnetic fluxφ for the hexagonal dot, as shown in Fig. 6. We see thatthe Landau levels divide the whole parameter space of (E,φ) into different regions with behaviors ranging from regular,parallel line patterns to complicated irregular patterns.28 Wehave analyzed the case that the Fermi energy is below the firstLandau level, where the edge states recur with the period Eq ,leading to regular conductance oscillations of the same energyperiod. For EF > E1, there are two sets of edge states, leadingto two uncorrelated repetitive patterns, each with its own periodEq . This is also manifested in Fig. 6 for the hexagonal dotthat, in region 2 (between the first and the second Landaulevels), there are two sets of conductance lines: one with thesame slope as in region 1 (the overlapped gray lines) and

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YING, HUANG, LAI, AND GREBOGI PHYSICAL REVIEW B 85, 245448 (2012)

another with a larger slope (brown lines) that originates in thisregion but persists in regions between higher Landau levels. Inregion 3 a new pattern appears, as indicated by the blue dashedlines in Fig. 6. The corresponding edge states are also shownin Fig. 6 for these typical line segments. We see that, for a fixedmagnetic flux, as the Fermi energy is increased across a Landaulevel, a new set of edge states appears, adding a new set of linesegments in the conductance plot. Since the energy period Eq

is uncorrelated for different types of edge states, as can be seenfrom Fig. 6, the conductance will fluctuate randomly whenthere are many sets of edge states. This explains the transitionfrom regular conductance oscillations to random conductancefluctuations, as shown in Figs. 2(a), 3(a), and 4(a). A similaranalysis can be carried out when the magnetic flux is varied[Figs. 2(b), 3(b), and 4(b)]. Since the transition is caused bythe crossing of Landau levels and the variation of the edgestates, it holds regardless of the detailed geometric shape ofthe quantum dot and the nature of the underlying classicaldynamics, i.e., integrable or chaotic. The transition can thusbe characterized as universal.

While our discussion has been focused on the hexagonaldot, here we briefly show that the same mechanism leading toregular conductance oscillations and the transition to randomfluctuations holds for other geometries as well. To demonstratethis in a comprehensive manner, we show in Fig. 7 theconductance in the (φ,E) plane for all three cases. We see

FIG. 7. (Color online) (a)–(c) Conductance G(E,φ) for thehexagonal-, square-, and stadium-shaped graphene quantum dots,respectively, where the colors indicate the values of the conductance.

that the conductance is symmetric with respect to reversal ofthe magnetic flux [T (φ) = T (−φ)] due to the two-terminalcharacteristic of our device.18 The patterns of the conductanceoscillations and fluctuations for the three cases are apparentlysimilar, due to the fact that the patterns are all partitioned bythe Landau levels [e.g., Eq. (6)] that do not depend on thegeometric details of the device. However, the fine structurescan be different. First, below the first Landau level, the slopesof the line patterns indicate the size of the device becausethe edge states are exactly circulating the “edge” of thedevice (Fig. 6), which are slightly different for the threecases. Second, above the first Landau level, the details ofthe conductance patterns are more distinct. This is because,in contrast to the edge states below the first Landau level,these states are now more dispersive and also depend on theshape of the device (comparing the LDS patterns in Figs. 2–4).Third, conductance fluctuations in the chaotic stadium billiardtend to be more smooth as compared with those in the twointegrable cases.29,30 This feature is especially pronounced inthe small-φ regime. When the classical dynamics is chaotic,the characteristic energy scale in the conductance-fluctuationpattern of the underlying quantum dot tends to be muchlarger,11 leading to a smoother variation. For quantum dotswith integrable or mixed dynamics, there are sharp resonancesin the conductance-fluctuation curves. This can be seen, e.g.,from the sudden change of the color scale from blue to red,or vice versa, in Fig. 7(b) for φ ∼ 0. [In the chaotic case, thechange in the color scale is much more smooth, as shown inFig. 7(c)]. In addition, in the chaotic graphene quantum dot,there is level repelling, which can also be seen from Fig. 7(c)in the φ ∼ 0 regime, where the conductance lines tend to avoideach other, a feature that is absent in both Figs. 7(a) and 7(b).

V. CONCLUSION

Previous works on conductance fluctuations associatedwith transport through nanoscale, quantum-dot systemsemphasized the difference between situations where theunderlying classical dynamics are chaotic or integrable.1–4

A general understanding is that Fano-type31 of sharpresonances typically occur in dot systems with integrableclassical dynamics, and chaos can effectively smooth out theseresonances quantum-mechanically. This picture holds for both2DEG and graphene systems in which the quantum dynamicsare nonrelativistic and can be relativistic, respectively, andit has been suggested recently32 that altering classical chaoscan effectively modulate quantum transport in terms ofconductance-fluctuation patterns.

We find that the presence of magnetic field can alterthe existing understanding of the quantum manifestations ofclassical chaos in that the difference in the quantum transport ascaused by different types of classical dynamics can diminish.As a result, universal behaviors emerge. The remarkablephenomenon has been observed in graphene quantum dotsof integrable and chaotic geometries. In particular, the con-ductance curves contain both regular oscillations and randomfluctuations, and the transition is caused by the emergence ofnew edge states when crossing the Landau levels. In the regionof regular oscillation, the periods in the Fermi energy and inmagnetic flux are related to the size of the device in a universal

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manner, regardless of the nature of the corresponding classicaldynamics. The key to this universal scaling is the quantizationof classically circulating edge orbits, which does not dependon the specific details of the geometrical shape of the dot.The details do appear in the fine-scale variations, where therandom conductance fluctuations are typically smoother whenthe classical dynamics is chaotic.

ACKNOWLEDGMENTS

This work was supported by AFOSR under Grant No.FA9550-12-1-0095 and by ONR under Grant No. N00014-08-1-0627. L.H. was also supported by NSFC under GrantNo. 11005053 and by the Fundamental Research Funds forthe Central Universities.

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